Calculating the Perpendicular Slope

Find the Equation of a Perpendicular Line

When working with lines in mathematics, understanding perpendicular lines is crucial. A perpendicular line is a line that intersects another line at a right angle (90 degrees). In this article, we will explore how to find the equation of a perpendicular line.

Understanding the Concept

To find the equation of a perpendicular line, we need two essential pieces of information: the slope of the original line and a point that lies on the perpendicular line. Let's denote the slope of the original line as m and a point on the perpendicular line as (x₁, y₁).

Calculating the Perpendicular Slope

The slope of a perpendicular line can be determined by taking the negative reciprocal of the slope of the original line. We can calculate it using the formula:

m_{perpendicular} = -1/m

Finding the Equation

Once we have the perpendicular slope and a point on the line, we can use the point-slope form of a line to find the equation of the perpendicular line. The point-slope form is given by:

y - y₁ = m_{perpendicular}(x - x₁)

Substitute the values of m_{perpendicular} and (x₁, y₁) into the equation to get the final equation of the perpendicular line.

Example

Given the equation of the original line: y = 2x - 3

To find the equation of a line perpendicular to this line, we need to find the negative reciprocal of the slope.

Step 1:

Calculate the slope of the original line. The slope of the original line is 2.

Step 2: Find the negative reciprocal of the slope. The negative reciprocal of 2 is -1/2.

Step 3: Write the equation of the perpendicular line using the negative reciprocal slope and a point on the line.

Let's assume a point on the line, for example, (1, 4). Using the point-slope form, we have:

y - y1 = m(x - x1)

y - 4 = -1/2(x - 1)

y - 4 = -1/2x + 1/2

y - 4 = -1/2x + 1/2

y = -1/2x + 9/2

Conclusion

Understanding how to find the equation of a perpendicular line is an important skill in mathematics. By determining the perpendicular slope and using the point-slope form, we can easily derive the equation. Remember to calculate the negative reciprocal of the original slope and substitute the values of the point to get the equation. Practice with more examples to strengthen your understanding of perpendicular lines.

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